My interest in 5 string is to use a high C for chord work.
Almost nothing comes ex factory E-C, so you're going to need a set up and maybe a new nut.
I love my Ibanezes and they are my go to recommendation. Physically any SR will fell like any other, it's just the materials and electronics that differ. New the 405 seems to be about the price/$ value point, but there are a bajillion second hand and you might be able to find a higher model for the same money as mew. My first SR 6 was the base SR256B and for what I paid for that new, I lucked onto a 1 owner near mint SR706 with factory HSC for just a bit more. I still have both.
I've tried a couple of Corts and they don't feel quite the same but are not too far off in terms of feel. Not enough experience with Schecters, but have been told the same.
WRT the Steinberger 'Woodbergers', they're good basses, but because of the small body and no pivot plate like the XL2, the reach to fret 1 is longer as the bass hangs further to the left. The fix costs $10 and a 1/2 hour of your time to make a strap extender. Good basses all around. I have 3 80's Hohners, a recent Spirit, Spirit guitar and an XL2 so I have some perspective. I'm so used to playing the XL2 that the small size feels at home, but to many it feels odd, and some people will make asinine remarks about it's appearance.
I have little experience with the full size body versions, but would expect them to sound the same, just have a different hang because of the shape. Double ball strings aren't hard to get, it's just you'll likely need to order them rather than stop by the local store unless you get a single ball adapter.
Because you want E-C, I'd say try 6s in the same range as you're looking at. For example, try an SR506 and like the tone and feel, then an SR505 will feel and sound the same just with a narrower neck and one less string.
One final suggestion is a Bass VI style instrument.
Harley Benton have one that looks excellent for the money with a video review
here. I have 2 Ibanez SRC6s which are the same concept and love them. Their main use for me at the moment (post lockdown anyway) is to accompany two girls who sing and play acoustics and I play the melody work under their rhythmic strumming.